### Hysteresis and Eddy Current Loss

1.Introduction
- When magnetic circuit are subjected to alternating magnetic field, There are two types of power losses in the magnetic core. These losses are significant in determining the heating, rating and efficiency of ac operated magnetic devices such as transformer and ac machine. The power losses are due to the hysteresis in the magnetic materials and eddy current step up in it due to the alternating field. Thus, the power loss due to hysteresis effect is called hysteresis losses and that due to eddy current is referred as eddy current loss. The two losses together are known as core or iron loss.
- Hysteresis and eddy current losses together constitute the no-load loss.

2. Theory
Fig. 1   Hysteresis loss
Eddy current loss, occurring on account of eddy currents produced due to induced voltages in laminations in response to an alternating flux, is proportional to the square of thickness of laminations, square of frequency and square of effective (r.m.s.) value of flux density.
Hysteresis loss is proportional to the area of hysteresis loop (figure 1(a)). Let e, io and φm denote the induced voltage, no-load current and core flux respectively. As per equation 1.1, voltage e leads the flux φm by 90°.
..........(1)
Due to hysteresis phenomenon, current io leads φm by a hysteresis angle (ß) as shown in figure 1(b). Energy, either supplied to the magnetic circuit or returned back by the magnetic circuit is given by

............(2)
- If we consider quadrant I of the hysteresis loop, the area OABCDO represents the energy supplied. Both induced voltage and current are positive for path AB. For path BD, the energy represented by the area BCD is returned back to the source since the voltage and current are having opposite signs giving a negative value of energy. Thus, for the quadrant I the area OABDO represents the energy loss; the area under hysteresis loop ABDEFIA represents the total energy loss termed as the hysteresis loss.
- This loss has a constant value per cycle meaning thereby that it is directly proportional to frequency (the higher the frequency (cycles/second), the higher is the loss). The non-sinusoidal current io can be resolved into two sinusoidal components: im in-phase with φm and ih in phase with e. The component ih represents the hysteresis loss. The eddy loss (Pe ) and hysteresis loss (Ph ) are thus given by
........(3)
........(4)
t : is thickness of individual lamination
kand k2 : are constants which depend on material
Brms  : is the rated effective flux density corresponding to the actual r.m.s. voltage on the sine wave basis
Bmp  : is the actual peak value of the flux density
n : is the Steinmetz constant having a value of 1.6 to 2.0 for hot rolled laminations and a value of more than 2.0 for cold rolled laminations due to use of higher operating flux density in them.

- In r.m.s. notations, when the hysteresis component (Ih) shown in figure 1(b) is added to the eddy current loss component, we get the total core loss current (Ic).
- In practice, the equations 3 and 4 are not used by designers for calculation of no load loss. There are at least two approaches generally used; in one approach the building factor for the entire core is derived based on the experimental/test data, whereas in the second approach the effect of corner weight is separately accounted by a factor based on the experimental/test data.

No load losses = Wt x Kb x w
or No load loss = (Wt - Wc) x w + Wc x w x Kc
Where :
w : is watts/kg for a particular operating peak flux density as given by lamination supplier (Epstein core loss),
Kb : is the building factor,
Wc : denotes corner weight out of total weight of Wt, and
Kc : is factor representing extra loss occurring at the corner joints (whose value is higher for smaller core diameters).